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Why is the product of negative numbers positive?

  • Last Updated : 29 Nov, 2021

Algebra is the branch of mathematics dealing with arithmetic operations and their associated symbols. The symbols are termed as variables that may take different values when subjected to different constraints. The variables are mostly denoted such as x, y, z, p, or q, which can be manipulated through different arithmetic operations of addition, subtraction, multiplication, and division, in order to compute the values. 

Negative numbers

A negative number corresponds to an integer value that is less than zero. A negative number is used to denote a loss or deficiency. Negative numbers are prepended with a negative sign. For instance, loans or credit are denoted with negative numbers

Rule: Opposite of an opposite is equivalent to the original value. 

For example, −(−3) = 3 

Why is the product of negative numbers positive?

Answer:

Upon multiplication of a negative number by another negative number, the resultant operation is positive in nature. 

To Prove: The product of two negative numbers or terms is positive:

(−a)(−b) = ab

where, a and b can be:

  • Numbers (i.e. a = 5, b = 1/2)
  • Constants
  • Variables
  • Expressions [i.e. a = (y2 + 6), b = (h − w + z)]

Proof

To prove (−a)(−b) = ab, we can consider the equation:

x = ab + (−a)(b) + (−a)(−b)

It can be easily shown that x = ab and x = (−a)(−b).

Factor out −a

First, factoring out −a from the expression (−a)(b) + (−a)(−b):

x = ab +(−a)(b) + (−a)(−b)

Thus, we obtain,

x = ab + (−a)[b + (−b)]

Since, b + (−b) = 0

x = ab + (−a)(0)

Thus,

x = ab

Factor out b 

Factoring out b from the expression ab + (−a)(b):

x = ab + (−a)(b) + (−a)(−b)

x = b[a + (−a)] + (−a)(−b)

x = b(0) + (−a)(−b)

Therefore,

x = (−a)(−b)

Result

Since x = ab and x = (−a)(−b):

(−a)(−b) = ab

This can be extended to any even amount of negative numbers by factoring out in steps:

(−a)(−b)(−c)(−d) = ab(−c)(−d) = abcd

Summary

The method easily proves (−a)(−b) = ab.

The fact that the product of two negative numbers, terms, or expressions is positive can be extended to any even number of negative items.

Sample Questions

Question 1. Find the product of -3a × -20b

Solution:

Here we have to find the product of -3a × -20b

As we know that when we multiply two negative numbers the answer will be positive

Now multiplying

-3a × -20b = +60ab

Therefore, the solution is positive.

Question 2. Evaluate 2ab + (−a)(5b) + (−2a)(−3b)?

Solution:

Here we have to find the product of

= 2ab + (−a)(5b) + (−2a)(−3b)

First solving the brackets

= 2ab + (-a × 5b) + (-2a × -3b)

= 2ab + (-5ab) + 6ab

= 2ab – 5ab + 6ab

= 8ab – 5ab

= 3ab

Question 3. Evaluate  10ab + (−21a)[5b + (−10b)]?

Solution:

Here we have to find the product of 10ab + (−21a)[5b + (−10b)]

First solving the brackets

= 10ab + (−21a)× [5b −10b]

= 10ab – 21a × (-5b)

= 10ab + 105ab

= 115ab

Question 4. Find the product of {-(4x + 5x) × (12x – 16x)}?

Solution:

Here we have to find the product of {-(4x + 5x) × (12x – 16x)}

First solving the brackets

= {(-4x – 5x) × (12x – 16x)}

= (-9x) × (-4x)

= 36x2

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